Solving F(x) = 6 For F(x) = 2|x+6| - 4 A Step-by-Step Guide
Introduction to Absolute Value Functions
In mathematics, absolute value functions play a crucial role in various problem-solving scenarios. These functions, characterized by their V-shaped graphs, introduce an interesting twist due to the absolute value operator, which always returns a non-negative value. To truly grasp the concept, we first need to understand the basics of absolute value functions. The absolute value of a number is its distance from zero on the number line. So, the absolute value of a number is always non-negative. For example, |5| = 5 and |-5| = 5. This property leads to the V-shaped graph, where the function's behavior changes at the point where the expression inside the absolute value becomes zero.
Consider the general form of an absolute value function: f(x) = a|x - h| + k. Here, 'a' determines the steepness and direction of the V, 'h' represents the horizontal shift, and 'k' represents the vertical shift. Understanding these parameters is essential for analyzing and solving equations involving absolute value functions. We will delve into the specifics of solving for x when f(x) equals a certain value, using the given function as an example. This involves breaking down the problem into cases based on the sign of the expression inside the absolute value and carefully solving each case. Mastering this technique is invaluable for various mathematical applications and problem-solving contexts.
Problem Statement: f(x) = 2|x+6| - 4 = 6
The problem at hand involves finding the specific values of x for which the function f(x) = 2|x+6| - 4 equals 6. This requires us to solve the equation 2|x+6| - 4 = 6. Solving equations with absolute values necessitates a careful approach, as the absolute value introduces two possible scenarios. To solve an absolute value equation, we must consider both the positive and negative cases of the expression inside the absolute value. This is because the absolute value of both a positive number and its negative counterpart are the same. In this particular case, the expression inside the absolute value is (x+6). Therefore, we need to examine two distinct scenarios: one where (x+6) is positive or zero, and another where (x+6) is negative. By addressing both scenarios, we can ensure that we capture all possible solutions for x that satisfy the original equation. This methodical approach is crucial for accurately solving absolute value equations and understanding the behavior of absolute value functions. This type of problem is fundamental in algebra and provides a solid foundation for more advanced mathematical concepts.
Step-by-Step Solution
To effectively solve the equation f(x) = 2|x+6| - 4 = 6, we need to break it down into manageable steps. The first step involves isolating the absolute value term. This means performing algebraic manipulations to get the term |x+6| by itself on one side of the equation. Adding 4 to both sides of the equation gives us 2|x+6| = 10. Next, we divide both sides by 2, resulting in |x+6| = 5. This isolation step is crucial because it sets the stage for the next critical phase: considering both positive and negative cases. Once the absolute value term is isolated, we can address the two possible scenarios arising from the absolute value: the expression inside the absolute value is either equal to 5 or equal to -5. This branching out into two cases is the essence of solving absolute value equations. Each case will lead to a different linear equation, which we can solve using standard algebraic techniques. By methodically addressing each step, we can arrive at the correct solutions for x. This process not only provides the answers but also enhances understanding of how absolute values affect equation solving.
Case 1: x + 6 = 5
In the first case, we consider the scenario where the expression inside the absolute value, (x + 6), is equal to 5. This gives us the equation x + 6 = 5. To solve for x, we need to isolate x on one side of the equation. This can be achieved by subtracting 6 from both sides of the equation. Subtracting 6 from both sides, we get x = 5 - 6, which simplifies to x = -1. This value of x is a potential solution to the original equation, but it's crucial to verify it later. This case represents one possibility where the absolute value of (x + 6) is 5, specifically when (x + 6) is positive. Solving this linear equation is straightforward and demonstrates the basic algebraic principles involved in handling absolute value equations. By carefully applying these principles, we can systematically find all possible solutions.
Case 2: x + 6 = -5
In the second case, we consider the scenario where the expression inside the absolute value, (x + 6), is equal to -5. This gives us the equation x + 6 = -5. Similar to the previous case, we need to isolate x to find its value. To do this, we subtract 6 from both sides of the equation. Subtracting 6 from both sides, we get x = -5 - 6, which simplifies to x = -11. This provides us with another potential solution for x. This case addresses the situation where the absolute value of (x + 6) is 5, but (x + 6) itself is negative. The negative case is just as important as the positive case when dealing with absolute values because the absolute value function returns the magnitude of the expression, disregarding its sign. Solving this case involves the same algebraic principles as the first case, reinforcing the systematic approach required for solving absolute value equations. Both cases together provide a complete picture of the possible values of x that satisfy the original equation.
Verification of Solutions
After finding the potential solutions x = -1 and x = -11, it's essential to verify that these values indeed satisfy the original equation, f(x) = 2|x+6| - 4 = 6. This verification step is a crucial part of the problem-solving process, as it ensures that we haven't made any errors in our calculations and that the solutions are valid. We will substitute each potential solution back into the original equation and check if the equation holds true. This process not only confirms the correctness of the solutions but also reinforces our understanding of the function and its behavior. By performing this step, we can confidently present the final answer, knowing that it has been thoroughly checked and validated. Skipping the verification step can lead to incorrect answers, especially in more complex problems.
Verifying x = -1
To verify x = -1, we substitute this value into the original equation f(x) = 2|x+6| - 4 = 6. Replacing x with -1, we get f(-1) = 2|-1+6| - 4. Simplifying the expression inside the absolute value, we have |-1+6| = |5| = 5. So, f(-1) = 2(5) - 4. Further simplifying, we get f(-1) = 10 - 4, which equals 6. Since f(-1) = 6, the value x = -1 satisfies the original equation. This confirms that x = -1 is a valid solution. The process of substituting and simplifying demonstrates how the function behaves for this particular value of x. This verification step is not just about checking the answer; it's also about understanding the mathematical reasoning behind the solution. Each step in the simplification process contributes to a deeper understanding of the function and its properties. This thorough approach ensures accuracy and enhances comprehension.
Verifying x = -11
Next, we verify the solution x = -11 by substituting it into the original equation f(x) = 2|x+6| - 4 = 6. Replacing x with -11, we have f(-11) = 2|-11+6| - 4. Simplifying the expression inside the absolute value, we get |-11+6| = |-5| = 5. Therefore, f(-11) = 2(5) - 4. Continuing to simplify, we find f(-11) = 10 - 4, which equals 6. Since f(-11) = 6, the value x = -11 also satisfies the original equation. This confirms that x = -11 is a valid solution. Just like with the previous solution, this verification step is essential to ensure the correctness of our work. It demonstrates that the negative case we considered earlier also leads to a valid solution. The consistent result from both verifications reinforces the systematic approach we used to solve the problem. This comprehensive verification process leaves no room for doubt and provides a strong foundation for the final answer.
Final Answer
After solving the equation f(x) = 2|x+6| - 4 = 6 and verifying the solutions, we can confidently state the final answer. The values of x that satisfy the equation are x = -1 and x = -11. These two values are the only points where the function f(x) equals 6. This final answer encapsulates the entire problem-solving process, from the initial analysis of the equation to the careful verification of the solutions. Presenting the final answer clearly and concisely is crucial for effective communication in mathematics. The answer not only provides the numerical values but also represents the culmination of a logical and systematic approach. This comprehensive solution demonstrates a deep understanding of absolute value functions and equation-solving techniques. The process followed here can be applied to a wide range of similar problems, making it a valuable skill in mathematics.
Therefore, the solution set is {-11, -1}.
